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        Space (mathematics)
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The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.


In the ancient mathematics, "space" was a geometric abstraction of the
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
three-dimensional space observed in the everyday life. Axiomatization
of this space, started by Euclid, was finished in the 19
century. Non-equivalent axiomatic systems appeared in the same 19
century: the hyperbolic geometry (Nikolai Lobachevskii, Janos Bolyai,
Carl Gauss) and the elliptic geometry (Georg Riemann). Thus, different
three-dimensional spaces appeared: Euclidean, hyperbolic and
elliptic. These are symmetric spaces; a symmetric space looks the same
around every point.
 
Much more general, not necessarily symmetric spaces were introduced in
1854 by Riemann, to be used by Albert Einstein in 1916 as a foundation
of his general theory of relativity. An Einstein space looks
differently around different points, because its geometry is
influenced by matter.
 
In 1872 the Erlangen program by Felix Klein proclaimed various kinds
of geometry corresponding to various transformation groups. Thus, new
kinds of symmetric spaces appeared: metric, affine, projective (and
some others).
 
The distinction between Euclidean, hyperbolic and elliptic spaces is
not similar to the distinction between metric, affine and projective
spaces. In the latter case one wonders, which questions apply, in the
former --- which answers hold. For example, the question about the sum
of the three angles of a triangle: is it equal to 180 degrees, or
less, or more? In Euclidean space the answer is "equal", in hyperbolic
space --- "less"; in elliptic space --- "more". However, this question
does not apply to an affine or projective space, since the notion of
angle is not defined in such spaces.
 
The classical Euclidean space is of course three-dimensional. However, the modern theory defines an <math>n</math>–dimensional Euclidean space as an affine space over an <math>n<\math>–dimensional inner product space (over <math>\mathbb R<\math>); for <math>n=3</math> it is equivalent to the classical theory.
 
Euclidean axioms leave no freedom, they determine uniquely all
geometric properties of the space. More exactly: all three-dimensional
Euclidean spaces are mutually isomorphic. In this sense we have "the"
three-dimensional Euclidean space. Three-dimensional symmetric
hyperbolic (or elliptic) spaces differ by a single parameter, the
curvature. The definition of a Riemann space leaves a huge freedom,
more than a finite number of numeric parameters. On the other hand,
all affine (or projective) spaces are mutually isomorphic, provided
that they are three-dimensional (or n-dimensional for a given n) and
over the reals (or another given field of scalars).
 
Nowadays mathematics uses a wide assortment of spaces. Many of them
are quite far from the ancient geometry. Here is a rough and
incomplete classification according to the applicable questions
(rather than answers). First, some basic classes.
 
{| class="wikitable"
|-
! Space
! Stipulates
|-
| Projective
| Straight lines.
|-
| Topological
| Convergence, continuity. Open sets, closed sets.
|}
 
Distances between points are defined in metric spaces. In addition,
all questions applicable to topological spaces apply also to metric
spaces, since each metric space "downgrades" to the corresponding
topological space. Such relations between classes of spaces are shown
below.
 
{| class="wikitable"
|-
! Space
! Is richer than
! Stipulates
|-
| Affine
| Projective
| Parallel lines
|-
| Linear
| Affine
| Origin. Vectors.
|-
| Linear topological
| Linear space. Topological space.
|-
| Metric
| Topological space.
| Distances.
|-
| Normed
| Linear topological space. Metric space.
|-
| Inner product
| Normed space.
| Angles.
|-
| Euclidean
| Affine. Metric.
| Angles.
|}
A finer classification uses answers to some (applicable) questions.
 
{| class="wikitable"
|-
! Space
! Special cases
! Properties
|-
| Linear
| three-dimensional
| Basis of 3 vectors
|-
|
| finite-dimensional
| A finite basis
|-
| Metric
| complete
| All Cauchy sequences converge
|-
| Topological
| compact
| Every open covering has a finite subcovering
|-
|
| connected
| Only trivial open-and-closed sets
|-
| Normed
| Banach
| Complete
|-
| Inner product
| Hilbert
| Complete
|}
 
Waiving distances and angles while retaining volumes (of geometric bodies) one moves toward measure theory and the corresponding spaces listed below. Besides the volume, a measure generalizes area, length, mass (or charge) distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory.
 
{| class="wikitable"
|-
! Space
! Stipulates
|-
| Measurable
| Measurable sets and functions.
|-
| Measure
| Measures and integrals.
|}
 
Measure space is richer than measurable space. Also, Euclidean space is richer than measure space.
 
{| class="wikitable"
|-
! Space
! Special cases
! Properties
|-
| Measurable
| standard
| Isomorphic to a Polish space with the Borel σ-algebra.
|-
| Measure
| standard
| Isomorphic <i>mod</i> 0 to a Polish space with a finite Borel measure.
|-
| Probability
| The whole space is of measure 1.
|}

Latest revision as of 03:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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